Math 131C : Basic exam "bootcamp"

• Course description: Review of undergraduate topics in linear algebra, analysis, and applied mathematics for the UCLA Basic Examination.
• ANNOUNCEMENTS: None yet.

• Instructor: Zubin Gautam, MS 6603
• Lecture: T 11:00 - 11:50, R 11:00 - 12:50 MS 6118
• Office Hours: M 12:00 - 12:50, R 5:15 - 6:05
• Grading scheme: Some combination of homework, presented problems, and exam(s) to be determined.
• Recommended texts (optional):
  Linear algebra:
  • Peter Petersen's lecture notes (http://www.math.ucla.edu/~petersen)
  • L. Smith, Linear Algebra
  • S. Lang, Linear Algebra
  • S. Friedberg, A. Insel, and L. Spence, Linear Algebra
  Analysis and advanced calculus:
  • T. Gamelin and R. Greene, Introduction to Topology
  • W. Rudin, Principles of Mathematical Analysis
  • T. Tao, Analysis I and Analysis II
  • C.H. Edwards, Advanced Calculus of Several Variables

Homework assignments

• Homework 1, due Tuesday, 1/15/2008.
• Homework 2, due Friday, 1/25/2008.
• Homework 3, due Friday, 2/1/2008.
• Homework 4, due Friday, 2/8/2008.

Topics covered

• Week 1: Basic definitions of linear algebra (vector spaces over fields, subspaces, spanning, linear independence, bases), linear transformations, matrix representations of linear transformations, vector space/algebra isomorphism of spaces of linear transformations with appropriate spaces of matrices, the Rank-Nullity Theorem. Sample Basic problems: F05#9, W02#8.
• Tuesday, 1/15: Definition of a metric space, open and closed sets, sequences and limit points, Cauchy sequences and completeness, compactness, equivalent formulations of compactness in metric spaces (open cover definition, Bolzano-Weierstrass (sequential compactness), Heine-Borel (complete and totally bounded)). Sample Basic problem: F01#1.
• Thursday, 1/17: Basic definitions of inner product space theory, orthogonality, orthonormal bases and the Gram-Schmidt construction, transposes/adjoints of matrices, adjoint operators, symmetric/self-adjoint matrices and operators, orthogonal/unitary matrices and operators, dual spaces, canonical dual basis, reflexivity, annihilator of a subspace, transpose of a linear transformation, finite-dimensional Fredholm Alternative (dual space and inner product space formulations).
• Tuesday, 1/22: Eigenvalues and eigenvectors, characteristic polynomial, the Spectral Theorem. Sample Basic problems: F01#10, S05LA#3
• Thurday, 1/24: Equicontinuity, the Lebesgue covering lemma, the Arzelà-Ascoli Theorem. Sample Basic problems: S05#6, F03#2.
• Tuesday, 1/29: The Stone-Weierstrass Theorem. Sample Basic problem: F07#5a
• Thursday, 1/31: Riemann integration. Sample Basic problems: F07#5b, F07#11, F07#12
• Tuesday, 2/5: Taylor's Theorem with remainder, Fundamental Theorem of Calculus, numerical integration (midpoint rule). Sample Basic problem: F07#4.
• Thursday, 2/7: Fundamental Theorem of Calculus (part 2), differentiation under the integral sign, Banach Fixed Point Theorem (contraction mapping principle), equivalence of norms on finite-dimensional vector spaces. Sample Basic problems: S05A#1, S05A#2, S04#6.
• Tuesday, 2/12: Vector-valued functions of several variables, the derivative as a linear transformation, Jacobian matrix, partial derivatives, directional derivatives, equality of mixed partials.
• Thursday, 2/14: Mean Value Theorem and Taylor's Theorem (multivariable setting), Inverse Mapping Theorem. Sample Basic problem: F03#6.
• Tuesday, 2/26: Implicit Mapping Theorem.
• Thursday, 2/28: Minimal polynomial of a linear transformation/matrix, rational canonical form, Jordan canonical form.
• Tuesday, 3/4: Matrix exponential, application to constant-coefficient systems of linear ordinary differential equations.

Files

• Notes on inner product spaces and spectral theory.
• Solution to Fall 2003 #7.